The study of
multiple ergodic averages was initiated in the seminal work of Furstenberg [1]}, where an ergodic theoretic proof of Szemerédi's theorem on arithmetic progressions was given. Since then a variety of multiple ergodic averages has been studied, resulting in
new and far reaching combinatorial consequences. A rather general family of problems is as follows: We are given
a collection of integer sequences \(a_1,\ldots , a_\ell \colon {\mathbb {N}}\rightarrow {\mathbb {Z}}\) , and we are asked, for an arbitrary invertible measure preserving system \((X,\mu ,T)\)  and functions \(f_1,\ldots , f_\ell \in L^\infty (\mu )\) , to understand the limiting behavior in \(L^2(\mu )\)  of the averages
(with \({\mathbb {E}}_{n\in [N]}\)  we denote the average \(\frac{1}{N}\sum _{n=1}^N\) )
\({\mathbb {E}}_{n\in [N]} \, T^{a_1(n)}f_1\cdot \ldots \cdot T^{a_\ell (n)}f_\ell \) 
